Interval cycle

In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class.[1] In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached (e.g. starting from C, going up by 3 semitones repeatedly until eventually C is again reached - the cycle is the collection of all the notes met on the way). In other words, interval cycles "unfold a single recurrent interval in a series that closes with a return to the initial pitch class". See: wikt:cycle.

Twelve-tone interval cycles[1] complete the aggregate: C1 once (top) or C6 six times (bottom).

Interval cycles assume the use of equal temperament and may not work in other systems such as just intonation. For example, if the C4 interval cycle used justly-tuned major thirds it would fall flat of an octave return by an interval known as the diesis. Put another way, a major third above G♯ is B♯, which is only enharmonically the same as C in systems such as equal temperament, in which the diesis has been tempered out.

"Similarly," to any pair of transpositionally related sets being reducible to two transpositionally related representations of the chromatic scale, "the pitch-class relations between any pair of inversionally related sets is reducible to the pitch-class relations between two inversionally related representations of the semitonal scale."[3] Thus an interval cycle or pair of cycles may be reducible to a representation of the chromatic scale.

As such, interval cycles may be differentiated as ascending or descending, with, "the ascending form of the semitonal scale [called] a 'P cycle' and the descending form [called] an 'I cycle'," while, "inversionally related dyads [are called] 'P/I' dyads."[4] P/I dyads will always share a sum of complementation. Cyclic sets are those, "sets whose alternate elements unfold complementary cycles of a single interval,"[5] that is an ascending and descending cycle: